Problem: A battery was charged. When the charging began, it was $23$ percent full. After $30$ minutes of charging, the battery was $89$ percent full. How fast was the battery charged?
Solution: Let's say that the percentage of the battery's capacity that is charged increases by $V$ percent each minute. Then, it increases by $V\cdot T$ percent in $T$ minutes. In addition, we know that when the charging began, the battery was $23$ percent full. The percentage of the battery's capacity that was charged at any given time is found by taking the percentage the battery was already charged when the charging began and adding to it the percentage that was additionally charged since then. We can express this with the equation $C=23+V\cdot T$, where: $C$ represents the percentage of the battery's capacity that is charged at a given time $V$ represents the charging rate (in percent per minute) $T$ represents the time (in minutes) We know that after $30$ minutes $(T={30})$, the battery was $89$ percent full $(C={89})$. Let's plug these values into the equation to find the value of $V$. $ \begin{aligned}{89}&=23+V\cdot{30}\\ 30V&=66\\ V&=2.2\end{aligned}$ Therefore, the battery was charged at a rate of $2.2$ percent per minute. To find how long it took the battery to be fully charged, we can plug $C=100$ into the equation and solve for $T$. $ \begin{aligned}100&=23+2.2T\\ 2.2T&=77\\ T&=35\end{aligned}$ The battery was charged at a rate of $2.2$ percent per minute. It took the battery $35$ minutes to be fully charged.